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Polynomial Identities of Hopf Algebras
by
© Mikha.il
v.
KotchetovA the.5i$ submitted to the School of Graduate Studies
inpartial fulfilment of the requirements!or the degree of
DoctorQj Philosophy
Department of Mathematics and Statistics Memorial University of Newfoundland
September 2002
St.John's Newfoundland Canada
Abstract
In this dissertation we consider Hopf algebras that satisfy a polynomial iden- tity as algebras or coalgebras. The notion of a polynomial identity for an algebra is classical. The dual notion of an identity for acoalgeurais new.
In Chapter 0 we give basic definitions and facts that are used throughout the rest of this work
Chapter 1 is devoted to coalgcbras with a polynomial identity. First we introduce the notion of identity of a coalgebra and discuss its general prop.
crties. Then we study what classes of coalgebras are varieties, i.e. call be definedbya set of identities. In the case of algebras, varieties are character- izedbythe classical Theorem of Birkhoff. Somewhat unexpectedly, the dual statement for coalgebras does not hold. Further, we give two realizations of a relatively (co)free coalgebra of a variety: one via the so called finite dual of a relatively free algebra and the other a direct construction using some kind of symmetric functions.
In Chapter 2 we give necessary and sufficient conditions for a cocommuta- tive Hopf algebra (with additional restrictions ill the case of prime character- istic) to satisfy a polynomial identity as an algebra. These results generalize the well-known Passman's Theorem on group algebras with a polynomial
ABSTRACT lii identity and Bahturin.Latysev's Theorem on universal enveloping algebras with a polynomial identity. The proofs for the case of prime characteristic are given ill Chapter 4.
Tn Chapter 3 we dualize the results of Chapter 2toobtain some criteria for a commutative Hopf algebra (assumed reduced in the case of prime char- acteristic) to satisfy an identity as a coalgebra. We also extend our result in charecteristic zero to a certain class of nearly commutative Hopf algebras (pscudoinvolutive Hopf algebras of Etingof-Celaki).
Finally, in Chapter 4 we usc the interpretation of cocommutative Hopf algebras as formal groups to prove the results of Chapter 2. Our method also demonstrates that Bahturin-Laty§ev's Theorem in characteristic zero is ill factII.corollary of Passman's Theorem
For the most part, this dissertation is based on my paper1'l [19], [20], and [211·
Acknowledgements
First of all, I would like to thank my supervisorProf,YuriA,Bahturin for his guidance during my studies at both Moscow State University and Memorial University of Newfoundland, \'\/ithout his advice, encomagement, aud finan- cial support this dissertation would not have been written. 1 am grateful to the Faculty of Mechanics and Mathematics of Moscow State University for my mathematical education. I also thank the Department of Mathematics and Statistics of Memorial University for providing a friendly atmosphere and facilities for my Ph.D. programme. In particular, I would like to thank Drs. E. Goodaire and H. Gaskill for their kind concern about my life and career.Iam also grateful to Dr. V. Petrogradsky for llseful discussions. I acknowledge the financial support of the School of Graduate Studies during my years at Memorial University. Finally,Ithank my parents and friends for their invaluable moral support.
iv
Contents
Abstract Acknowledgements Definitions and Basic Facts
0.1 Coalgebras and Comodules 0.2 Bialgebras andHope Algebras 0.3 Polynomial Identities 0.4 Some Topological Notions 0.5 QueFactfmlllDescent Tht->ory 1 Identities of Coalgebras
1.1 Coalgebras with a Polynomial Identity 1.2 FreeCoaJgebra.~
1.3 Varieties of Coalgebras 1.4 Relatively Free Coalgebras 1.5 Some Examples
2 PI Cocommutative Hopf Algebras
iv
16 18 22
2.
24 27 33 41 50 56
CONTENTS
2.1 Overview of Known Results 2.2 The Case of Zero Characteristic 2.3 The Case of Prime Characteristic 3 CPI Commutative Hopf Algebras
3.1 Preliminaries
3.2 The Dual Passman Theorem.
3.3 Pseudoinvolutive Hopf Algebras 4 Divided Powers and Power Series
4.1 Hyperalgebras and Formal Groups.
4.2 PI Coroouced Hyperalgebras Are Commutative 4.3 Smash Products with Divided Power Algebra Bibliography
, j
56 64 73 81 81 85 92 96 96 108 118 127
Definitions and Basic Facts
Throughout k will denote the ground field. All vector spaces, algebras, tensor products, etc. will be considered oyer k unless stated otherwise.kwill denote the algebraic closure of k.
All algebraswillbe assumed associative and unital, and algebra maps A)- tA2will be required to send the unit element ofAltothe noit element ofA2 .In particular, a subalgebra ofAmust contain the unit clement ofA.
Z will denote the set of integers and N the set of positive integers.
0.1 Coalgebras and Comodules
In this and the following section we refer to the excellent monograph of S.Montgomery [25] for the basic properties of coalgcbras and Hopf algebras.
See the bibliography of [25] for the references to original papers The notion of a coalgcbra is the dual of the notion of an algebra. We first express the associativity and unit axioms via commutative diagrams so that wecan dualize them.
Definition 0.1.1. Ak-algebm is a k-vectorspaceA together with two k- linear maps, multiplication m :A 0 A ---+ A and unit u :II:-+A, such that
DEFINITIONS AND BASIC FACTS [he following diagrams are commutative:
associativity A0A0A~A0A
id0mj
1m
A 0 A - -m- - A
unit
Definition 0.1.2. Ak-coalgcbrais a k-vector space C together withtwo k-linear maps, cOlnulti!,licatiolll.i. C-tC 0 Cami counitc .C-tk,such that the following diagrams are cOlllmutative:
coassociativity C - _ t . _ - C0C
t.1 I
t.0idC0C~C0C0C
counit k0C---!LC~C0k
'~tr'
C0CWe sayC iseocommutativeifde is a symmetric tensor fol' any e E C. A subspaceDeeisasubcoalgebmif6.DcD 0 D
Definition 0.1.3. Let C andDbe coalgebras, with comultiplicatiousde anddD,and counit.sccandCD,respectively.Alinear mapf :C-tDisa homomorphism of coalgebrasif6.1)0f
=
(f0f)06.candec=eo0f.A subspaceIe0 isacoidealifdle100+001ande(I) =0.It is easy to check tIlat ifI is a coideal, then the space OJ [ is a coalgebra with comultiplication induced fromd,and conversely.
DEFINITIONS AND BASIC FACTS
We will now see that there is a very close relationship between algebras and coalgebras, by looking at their dual spaces.IfV is a vector spacc, we willoften use the symmetric notation(f,v) instead of f(v), for vEV and fEV',
IfCis a coalgcbra, thcnC'is an algebra, with multiplicationm=~"
and unitu.
=
c'.IfCis oocommutatil'e, thenC"iscommutative.However, if we begin with an algebraA,then difficulties arise. For, ifA is not finite.dimensional, the image ofm" :A"--t(A0A)'does not have to be a subspace ofA'0 A". Thelargest subspace ofA'whose image lies in A'0A',is the so calledfillite dual·
AO= {JE.'1"1/(/)=0 for some idealI<lA, dimA/I<oo}.
AO is a ooalgebra with comultiplication~
=
m'andCOUllit£=
u'(restricted toA").IfAis cOllllllutative, thenA"is cocommutativeMoreover, the functor ()" is the right adjoint of ()., Le. for any algebra Aand coalgebra C, the sets of homomorphisms Alg(A,C')and Coalg(C, .'1°) arcinaone-to-one correspondence (see Lemma1.3.12)
Unfortunately,Nmay happen to be too small (even zero). The following conditionOilAisprecisely what weneedto guarantee that .'1°isbig enough to separate the elements ofA (in other words, AO is dense in A' in the sense of Definition0.4.1).
Definition0.1.4. AnalgebraAis calledresidually finite-dimensionalif its ideals of finite eodimensioll (i.e.J<lAwith dimA/!<00)intersect to 0 or, equivalently, for any0#a EA,there existsa finite-dimensional representa- tionrpofAsuch thatIf(a)#0
DEFINITIONS AND BASIC FACTS
The relationship between subcoalgebras, coideals, ideals, and subalgebras is the following.
Lemma 0.1.5. 1)LetCbe a coolgebra.
(a)AsubspaceDeCisa subcoa/gebraiff
0.1
=
{fEC'I(f,D)=
O}is an ideal ofC'.(b) A subspace [C Cisa evidealiff [1.
= {[
EC'I(f,I)=
O}isa subalgebra ofC'.2)LetA be an algebra (a) If BcAis a suba/gebm,
thenB.l ={fEAOIU,B)=O}is a coideal ofN (b) If IcAisanideal,
then[.l.={fE AOIU,1) =O} is asubcoalyebra ofAo. • Now we introduce the so calledsigma notationas follows. Let C be any coalgebra with comultiplication.6.: C-+C@C. For any c E C, we write:
.6.C=2.:>'-(I)@C(2j.
The subscripts (1) and (2)are symbolic, and do not indicate particular ele- mentsofC.
In sigma flotation, the coassociativity means that
so we simply write
L
c(l)181C{l)181C{3)=
.6.3c.Iterating this procedure gives, for allYn:::::2,DEFINITIONS AND BA.SIC FACTS
where~2
=
6. We will sometimes use the convention thatt::.[=
ideand 80=e.Nowwedualize the notion of a (unital) module by first writingitsdefi- nition in terms of commutative diagrams
Definition0.1.6.For a k-algebraA,a (left)A-modlileisa. k-spaceMwith a k-linear mapl' :A0At-tM such that the following diagrams commute:
Definition0.1.7.For a k-coalgebraG,a (right)G-comoduleis a k-space Mwitha k*linear mapp :M-tM181 C such that the following diagrams commute:
M - -P- - M 0 C
pi lid0~
M0C~M0C0C
A linear mapf :M-tNis ahomomorphism of (right) comodulesif it prcsen'csp:
A subspaceNCkfis asubcomoduleifp(N)CN0 C There is also sigma notation for right comodules: we write
p(m)=L1rl{o)0m(l) EMeG.
DEFINITIONS AND BASIC FACTS
Analogously, olle has left comodulcs, via a lIIaprJ :M--+C 0M,and we use the notation
p'(m)
= E
m(_l)03m(O)E C 0M,so that for both right and left comodules we havemoEMandTn(.)E Cfor i:;60.
The following Finiteness Theorem[25, Theorem 5.1.11 points out the main feature that distinguishes coalgebras and comodules frolll algebras and modules.
Theorem0.1.8.LeiCbe a coalgcbra.
1) AnyC-cQmodule 1\1 is locally finite in the sense that any finite subset of }.{ is contained in a finite·dimen.sional subcomodule.
2) Any finite subset ofC is wntained in a fillite-dimen.5ional subcoalgebra.
•
A nonzero coalgebra is calledsimpleif it does not have proper nonzero subcoalgebras. The theorem above implies thatallsimple coalgebras are finite-dimensionaL Italso implies that any nonzero coalgcbra has a simple subcoalgebra.
Definition0.1.9.Let G'bea coalgebra.
1) The coradicalcoradG' ofC is the slim or all simple subcoalgebras ofC.
2)CisC().5emisimpleif coracle
=
C.DEFINITIONSAND BASTC FACTS
3) C isirreducibleif coradC is simple or, equivalently,ifC contains only one simple subcoalgebra. Any maximal irreducible subcoalgebra ofC is called anirreducible component.
4) C ispointedif eyery simple subcoalgebra is one-dimensional 5) C isconnectedif coradC is one-dimensional
If D is a simple cocommutative coalgebra, thenD'is a simple (finite- dimensional) commutative algebra. It follows that auy cocommutative coal- gebra C over an algebraically closed field is pointed. A aile-dimensional suhcoalgebraDeeis necessarily of the form kg, where9E C isgroup- like: 09
t-
0 and Ao9=
90g.Distinct group-like elements of C are linearly independent, the set of all group-like clements is denotedO(C).Let us now quote for future reference the basic properties of irreducible components [25, Lemma 5.6.2 and Theorem 5.6.3J.
Lemma 0.1.10.Let C be a coalgebm
1) Any irreducible subcoalgebm ojCi8contained in a unique in"tducible component.
2) A sum oj distinct irreducible components i8 direct.
3) IjCis cocommut(ltive, thenC isthe direct sum oj its irreducible com-
ponents.
•
In fact, the coradical coraclC is the bottom piece of the so calledcomdical filtrationof C. We set Co
=
coraclC and for eac!! integern>0 define inductively·DEFINITIONS AND BASIC FACTS
Theil it turns out (see [25, Theorem 5.2.2]) that {Cn} is acoalgebra fil- trati011in thc following sense
f::,.C"Ct Ck 0Cn_k,
/;:",0
Cn C Cn+1>and C =
U
Cn.""
The conditions above guarantee that the spacecgr=$,,~oCn/C.._1
(withC~I=O)has a natural coalgebra structure.
Example 0.1.11. IfC isaconnected coalgebra, then Co is one-dimensional.
Itis spanJlcdbya groupo-like clement that we will dcnoteby 1(although there is no multiplication yet). LetP(C) be the set of a.111Jrimitive elements of C: x E C such that Ax = x 181 1+1 181 x. Then P(C) isa subspace and C1
=
klillP(C)[25,Lemma 5.3.2]The following lemma[25,Lemma5.3.4]shows that coraclC is the smallest piece a coalgebra filtration can start with.
Lemmu 0.1.12.LetCbeany colagebraand{Bn}n~oa coalgebra filtration ofC. Then Bo :) coraclC.
Corollary 0.1.13. If f: C-+Disa .mrjective coa/gcbra map,then f(coradC) :) coradD.
•
•
We conclude this section with another fundamental property of thecorad~
icalliltration [25, Theorem 5.3.1).
Theorem 0.1.14.LetCand D be coa/gebras andf ;C-+D a coalgebra map.Iffie.isinjective, thenfis injective. •
DEFINITIONS AND BASIC FACTS
Corollary 0.1.15.IIC is COnTlCCted and I:C~D is a coalgebm map such that flp(c)isinjective, thenfis illjective.
0.2 Bialgebras and Hopf Algebras
We now combine the notions of aJgebra and coaJgebra.
•
Deflnition 0.2.1. A k-S]MCeBis abiulgebraif(B,m,u)is an algebra, (B,~,£)is a coalgebra, and either of the following (!(juivalent conditions holds:~alldf.arc algebra morphisms or m anduarc coalgebra morphisllls.
Naturally, abialgebra homomorphismis a map which is both an algebra and a coalgebra homomorphism, and a subspaceDeBis asubbialgebraif it is both a subalgebra and a subcoalgebra. Similarly, a subspace 1 CBis a biidealif it is both an ideal and a coideal. The quotient BIIis a bialgebra precisely when 1 is a biidcal of B.
The last ingredient we need to define Hopf algebras is theconvolutioTI IJroduct.Namely, if C is a coalgebra andAis an algebra, then Homk(C,A) becomes an (associative) algebra undp.r the convolution:
for allI,gE HOIIlt.(C,A), c E C. The unit element of Homk(C,A)istl.Aocc.
Remark 0.2.2. Note that the multiplication on C' defined earlier is the salliC as the cOllvolution product Oil Homt(C,k)
=
C'.Deflnition 0.2.3. Let(H,m,u,~,£)be a bialgebra. ThenHis aHopi algebraif there exists an elemcntSE Hom(H,H)which is an inverse toidJl
DEFINITIONS AND BASIC FACTS under convolution, i.e.
Obviously, if suchS exists, itisunique.S is called the antipode of H 10
Naturally, a linear mapf :H--tKof Hopf algebras is aHopf homo- morphismif it is a bialgebra homomorphism andf(SlIh)
=
SKJ(h),for all hEN, A subspace D CHis a3ubHopfalgebmif it is a subbialgebra and SD cD. From the uniqueness of5it follows that if D C}fis a subbial·gebra that has its own antipodeSD,then D is in fact a subHopfalgebra and SD
=
Slu. A subspaceI cHisaHopf idealif it is a biideal andSICI, illthis situationHjlis a Hopf algebra wilh the structure induced fromH.The largest Hopf ideal is theaugmentationidealH+
=
Kere.Let us note that the antipode is necessarily an anti-algebra morphism, i.e
5(1) I,
S(,h) S(h)5(9), Vg,hE 11, and anti-coalgebra morphism, i.e.
'ISh)
L
(Sh)1" "(Sh)",«h),
L
Slh",) " 5(h1")' Vh E //.IfHis commutative or cocommutative, then 52=id. In general, S does not even have to be injective or surjective.
The basic examples of Hopf algebras are the following.
DEFINITIONS AND BASIC F'ACTS 11 1) The group algebra kG of any group G, witht.9
=
90g,£(9)=
1,Bg=g-Ifor aJly EG.
2) The universal enveloping algebraU(L) of any Lie algebra L, witht:.x
=
x 181 1+1 0x, c(x)
=
0,Sx=-x, for all x EL.3) The algebra of regular functionsO(G)on any affine algebraic gronp G, with.6. O(G)-+O{Gx G)
=
O(G)0G(G)corresponding to the group multiplication G x G-+G:(.6.J)(x,y)= f(xy), c{f)=
f(e), (5/)(x)=f(x-'),£0',IIfEO(G), x,yEG.The formCf two Hopf algebras are cocommutative, the latter is commu- tative. For any Hopf algebrafl,the setG(II) of all group-like clement!; is in fact a group (under the multiplication ofH),soHcontains the group algebrakG(H)(of course,G{H)may consist only of the unit element). Tile set P(H) of all primitive clements of H forms a Lie algebra under the com- mutator[x,
yJ
=xy - yx.The axioms of a bialgebra (or Hopf algebra) are self-dual. So it is not surprising that if(H,m,u,t.,f)is a bialgebra, then(H",t.',£',m',u·)is also a bialgebra, and ifHis a Hopf algebra with antipodeS,thenHeis a Hopf algebra with antipodeS'[25, Theorem 9,1.3J. We have to nse the finite dual[1"here rather than the wholeH',because comultiplication is not defined onfl'
Given a bialgebraHand a vector spaceV,we can tumVinto a "trivial"
(left) H-module by sel.ting for allItEH, v E V,
h·v=£{h)v. (0.2.1)
DEFINfTJONS AND BA.sIC FACTS 12 We can also turnVinto a "trivial" (right) H-comodule by setting for all vEV,
p(v) =v@1. (0.2.2)
Now ifVis any left H-modulc, the elements v EVthat satisfy (0.2.1), for allhEH,arc calledinvariants.Of course, a similar definition can be given for right modules. The set of all invariants is denoted byHVfor a left moduleVandVIIfor a right module.
IfVis a right H-comodule, the clements v EVthat satisfy (0.2.2) arc calledcoinvariunts,The set of all coinvariants is denoted by
v
coHfor right comodules andcoHVfor left comodulesFor any Hopf algebraH,the following actions and coactiolls ofHon itself are defined:
1)The left adjoint action'(adth)(k)
=
L:h{l)k(Sh(2)),for allh, kEH, 2)The right adjoint actioll:(ad.h}(k)=L:(Sh(l))kh(2),for allh, kEH, 3)The left adjoint coaction:PI : H-+H0H : h-+L:h(1)Sh(3)0h(2), 4)The right adjoint coaction: p.:H---j.H<81H:Ii---tEh(2)<81(Sh(1))h(3)' Definition 0.2.4. A subHopfalgebraKcHis callednorn~alif(adIH)KCKand(ad.f!)KCK.
AHopf idealI cHiscallednormalif PI(I)cH<81Iand P.(I) C 1€IH.
Obviously, all subHopfalgebras are normal for acommutati~'eH,and all Hopf ideals are normal for a cocommutativeH.
DEFINITIONS AND BASIC FACTS 13 There is a natural corresponclellce between normal subHopfalgebras and normal Hopf ideals ofHas follows.IfK CHisa normal subHop£algebra, thenH K+
=
K+ His a normal Hopf ideal. Conversely, ifIcHis a Hopf ideal, then we can considerfIasa rightHI l-comodule viaH-+H® HII: h-+L:h{l)13(h(2)+I),
and similarly on the left. Thus we eall define the spaces of right and left HI I-coinvariants inH: H"'JiIIandcollllH.IfI isa normal Hopf ideal, then colflf H= HcoHlfis a normal subHopfalgebra.Itis known that these two mappingsare inverse bijections in the case whenHis finite-dimensional or commutative or with cocommutative coradical (sec[25,Section3.4]).
Definition0.2.5.LetA be an algebra and H a Hopf algebra.
1)Ais a (left)H-module al.qebrajf it is a (left.) H-module such that h·1 f(h)1 aud
h·(ab) L:(h(l) . a)(h(2) .b), 'rIhEH, a,bEA 2) Ais a (right)H-colnodule algebraif it is a (right) H-comodule via
p:A-+A0Hsuch that
p(l) 1131 and
p(ab) L:Q.(o)b(o)13a(l)b(l). 'ria, bE A.
It is straightforward to verify that the adjoint action and coaction ofH on itself satisfy the above conditions.
We will needone more concept from general Hopf algebra theory, namely that of a crossed product.
DEFINITIONS AND BASIC FACTS 14 Definition 0.2.6. LetHbe a Hopf algebra andAan algebra. Assume that HmeasuresA,i.e. there is a linear mapHII}A-+A : h0a-+h .a such thatIi -I=c(h)l andII· (ab)
=
E(h(l)' a)(h(2)' b),for allhEH, a,bEA.Assume also thatu :H0H-+Ais a convolution-iuvertible map. The crossed product A#JH is A(9H as a vector space, with multiplical.ion
for allh, kEH,a,bEA,andwehave writteno.#hfor the tensora€Ih.
It is straightforward to derive the condiiions on anduso thatA#"H will he an associative algebra with the uuit 1#1 [25, Section 7.1]:
1) Ai.sa twisted H-module, i.e.1·a
=
a andfor allh,kE 11,aEA,
2)u is a (left) 2-cocycle, i.e.u(h,1)
=
u(l,h)= t(h)1and 2::[hl .u(k{l), m(I))]u(h(2)' k(2)m(2))= L
U(h(I)' k(l»)u(h(2)k(2)' m),(02.3) for allh,k,mEH.
Note that ifHis cocommutative andAcommutative (oruhas values in the centre ofA),thenAis simply an H-modllle algebra. Another special clLSe arises if we assumeutriviaLu(h, k)
=
t(h)t(k)l,for aJlh, kEH.Then againAis an H-modllie algebra, and the crossed productA#"Hwith such a a is called the smash product and denoted simply A#H.DEFINITIONS AND BASIC FACTS Iii The following decomposition theorem for cocommutativc pointed Hopf algebras is due to Kostant, Cartier, Gabriel, et al. and can be found in[25, Section5.61.
Theorem 0.2.7.Let H be a pointed cocommutative Hopf algebm overk. Let G
=
G(1I) be the group of group-like dements of 11 and HI the irreducible component of the simple subcoolgebrakl.TIIenGacts on H1byconjugation (whichi.'Jthe left adjoint action in this case) arid Hi.'Ji.'Jomorphic to the smash product H1#kG via h#g-.-fhg. Moreover, ifchark=0,then HI S! U(L), where L=
P(H)i.'Jthe Lie algebra of primitive elements. • Thus any pointed cocommutativc Hopf algebra can be represented as a smash product of a connected Hopf algebra and a group algebra, the former being just the universal envelope of a Lie algebra in the case of characteristic o.Remark0.2.8. The proof of the first statelllent of Theorem 0.2.7 given in [25, Section 5.6] does lIot require thatHbe cocommutative. It is sufficient to assume thatHis the sum of its irreducible components (this coudition is satisfied by any cocotnmutative Hopf algebra by Lemma 0.1.10).
We conclude this section by demonstrating the structure ofH-comodules ill the case }{=kG for some grollP G.Itis easy to see that, for any (right) kG-eomoduleV,we have
V
= ffi
Vg ,whereVg= {vEVIp(v)=
v0 g}, ,eosoV is a G-graded space. Conversely, any G-graded vector space can be turned into a kG-comodule by settingp(v)
=
v0gfor any homogeneousDEFINITIONS A.ND BASIC FACTS 16 vof degreeg.Moreover, kG-comodule algebras are equivalent to G-graded algebras in thiswa~'.
0.3 Polynomial Identities
Definition 0.3.1. Let A be an algebra overafield k (although most of the definitions and results of this ;;ection are still valid if k is a commutative ring with1). Let F(Xj, ... ,Xnl be a polynomial in n noncommuting variables with coefficients in k. We say thatAsatisfies the identityF
=
0 (or justF)F(aJ, ... ,anl=O, Val, ..,anEA.
An algebra A is called PI ifit satisfies the identity F
=
0 for some nonzero polynomialF.Because of the following theorem, multihomogeneous (i.e. homogeneous in each variable), and especially multilinear (i.e. linear in each variable), identities playa prominent role in the theory of polynomial identities. The proof of 1) is an easy exercise with Vandermonde's determinant, for 2) see e.g.[18,Section1.31.
Theorem 0.3.2. Let A be an algebra oller a field k and F a polynomial in noncommuting variables that is an identity for A.
1) Ifkis infinite, then every multillOmogeneous component of F is an identity for A.
2) The algebra A satisfies a multilinear identity of degree::;degF.More- over,ifk isa field of characteristic0,thenFi.5 equivalent to a (finite)
DEFINITIONS AND BASIC F.4.CTS 17 system {Fi }of multilinear identities,i.e. any algebra that satisfies F
mwt also satisfy all Fiand vice versa.
•
Corollary 0.3.3.If an algebraAwithlover an infinitefieldsatisfies an identity that does lIot follow from the commutativityXIX2 - X2X1> then A=O.
Thestandard polynomialof degree n is defined by sn(XI ,..,Xn)=L(-lYX"(l) .. X,,(,,),
lfES~
•
whereS" is the group of permutaiiolls and (-1)" is the sign of"Tr In partic- ular,82=XlX2~X2XI _
Since8"is multilinear and aliernating (i.e. vanishes upon any subsiitu- tionXi
=
Xjforii-j), any finite-dimensional algebra A will satisfy the standard identity s"=
0, for auy n>dimA. For example, the algebra M,,(k) of11xnmatrices satisfiesS,,'+I.This can be improved, as stated by the following classical Theorem of Amitsur-Kaplansky-LevitzkL Theorem 0.3.4.Tile matrix algebraM,,(k)satisfies the standard identity S:21\=
O.Itdoe.,~notsati~fyany nOlltrivial identity of degree<2n. •Finally, it is obvious that if an algebraAsatisfies a multihomogencous identityF,andBis any commutative algebra, thenA0BsatisfiesF. In particular,)"f,,(B)satisfies82",for any commutative algebraB.Italso fol- lows that ifAisPIandBis commutative, thenA0BisPI,The classical Theorem of A.Regev generalizes this simple observation: ifA and B arePI, thenA 08> BisPI(see e.g. [1]).
DEFINITIONS AND BASIC F.4.Cl'S
0.4 Some Topological Notions
18
Since we want to work overallarbitrary fieldk,we take k with the di.5crete topology,Le. all subsets of k are open, By a topological vector SlJace wewill mean a k-vector space endowed with a Hausdorff topology, with a funda- mental system of neighbourhoods of 0 consisting ofSUbSpace8,such that the additiOil ofv&tors is continuous. This is not the most general kind of a topo- logical ,'ector space, but it will be sufficient for our purposes. In particular, our definition forces any finite-dimensional v&tor space to have the discrete topology. By a topological algebra we will mean a topological vector space that is also a k-algebra such that the multiplication is continuous.
Recall that a partially ordered set I is called directed if for anyi,}E [ there existskEIsuch thati:oS kand} ::;k,A family{Z;}iElof clements of a topological spaceZis calledII.netifIis a directed set. A net{ZiliEI convergesto the pointzif for any neighbourhoodUofzthere existskE1 such thatZ;EUfor alli~k.A net{t';}iElin a topological vector space Vis caJled a Cau.chy net if for allY neighbourhood U of 0 there exists k E I such thatVi -VjEUas soon asi,j~k. A topological vector spaceVis completeif ally· Cauchy net converges to an element of V
Recall also the definilions of the direct and inverse limits. IfIis a directed set and {Z;}iEl isIifamily of sets endowed with a systp.m of mapstPij :Zi-+
Zj,for anyi:oSj, sllch thattP"
=
idz,andtP;k0tP,j==tP'k,then the direct limitis defined byI~Zi=
11
Z;/ "", iElthe quotient set of the disjoint union of Zi by the following equiva.lence re-
DEFINITIONS AND BASIC FACTS 19 lation:x ,...., yifxE Zi,Y E Zj, and tPik(X)
=
1{Jjk(Y) for somei,j::5 k.The inclusionsZj CUiEIZ; induce the canonical mapstjJj ;Zj-tl~Z;.
Clearly, we have'l/Jj 01/;ij
=
1/;i,andl~Z;, together with the mapstPi'is universal with respect to this property.Ifthe setsZi 1mve the structure of vector spaces, algebras, coalgebras, etc. that is preserved by the maps¢ij, then this structure is inherited byI~Zi.Dually, ifIis a directed set and{Z;}iEI is a family of sets endowed with a system of maps'Pij ZJ-tZi, forallyi ::5j,such that'Pii
=
idz• and r.pij0r.pjk =r.pik,then theinverse limit~ZicDZi
consists of all families{Zi}iE!ETIiE!Zisuch that!.piJ(Zj)
=
Zifor alli::5j The projections Il'EtZ;-tZj define the canonical maps'Pj :~Zi-+Zj.We have!.pij
°
'Pj=
/Pi,andI~Zi'with the maps'Pi,is universal with respect to this property.Ifthe setsZ; have the structure of vector spaces, algebras, etc. that is preservedbY'Pij ,thenI~Zi inherits this structure. Note that in general, the coalgebra structure isnot inherited because comnltiplication is not defined for an infinite direct product of coalgebras.IfZ; are topological spaces, thenI~lZi has a natural topology as a subset oflliEIZi.LetV be a topological vector space and suppose the subspacesVi, i E I, form a fundamental system of neighbourhoods of O. We writei::5j iff Ui:JUj. ThenIis a directed set and the inverse limit~VIUi of the discrete spaces VIUicontains Vasa topological subspace. Moreover, V is dense inl~VIUiandIe}VIUi is complete, soV=1~1VIUiis the completion ofV
DEFINITIONS AND BASIC FACTS 20 Definition 0.4.1. LetV be a vector space (without topology). then the dual vector spaceV'can be given the~*-weak"topologyu(V', V),i.e. the topology with a fundamental system of neighbourhoods of 0 of the form
U.".. ,••~(fEV'I (J, ",)~0, Vk~1, . . ,mj, where1)1, . .,urn EV,mE N
We immediately observe that all the setsV~l..~,.are subspaces of finite codimension, andV' iscomplete, thusV'is apro-finitevector space, i.e.
an inverse limit of finite-dimensional vector spaces. Conversely, every pro- finite topological vector spaceW has the form\I',whereV is the space of continuouslinear functions onW.Moreover, if'P:V-+\Vis a linear map, then 'P' :W·-+V' is a continuous linear map, and every continuous linear mapW'-+V'has the form 'P' for someIt :V--+IV (see e.g. [14, Sectioll
1.211·
If Vandi-Vare complete topological vector spaces, then\I1&1\Vcan be endowed with atensor product topologydefined by a fundamental system of neighbourhoodsofU of the formVII&IW +V0V2 ,whereVi CVandV2C IV are open subspaces. Hence we can define thecompleted ten80r product
V0W~,!':'((VjU;j" (WjU;)),
where{Vi} and{Vj}are fundamental systems of neighbourhoods of 0 inV andW,respectively.
If'P :V-+V' andt/J :W-+W' are continuous linear maps, we will denote by'P0t/J: V0W-+V'0W' the extension of 'P1&1t/J:V 0 W-+V' 0 W'.
DEFINITIONS AND BASIC FACTS 21 Now ifVand Ware vector spaces (without topology), andV'andW'are endowed with *-weak topology, thenV"('9W'is a dense topological subspace of the compkte space(V0W),.Therefore,(V181W)"=V"eW".
To conclude this section, let us introduce our main example of a topologi- cal algebra - the algebra of formal power series (in any number of variables).
But first we need to define the rnultiindex notation.
Definition 0.4.2.LetIbeaset. ArnultiindexonIis a map(t:I ...
{O,l, ... }such that
suppa={iEIla(i)#-O}
is finite, in otner words,0'EZ~),the direct sum of III copies of Z+. For any such awesct
lol~
L
o(i).For0',/3EZ~),we write a :5/3 ifo:{i}:5,8(i)for alii E 1.
For two lIlultiindiccs a,fJ, we define the combination number:
(0)
~n (o( i))
fl iE.uppa fJ{t)
if(3:50'and
°
otherwise. We also denote byE:ithe mllltiindex whose only nonzero component isE",{i)=1.Definition 0.4.3.Thealgebra of Jonnal power seriesk[[tiliE1]]is the topological vector spacenaEZ~)k (direct product of copies of k, with direct product topology), whose clements will be written as formal sums
DEFINITlONS AND BASTC FACTS with multiplication defined by the Cauchy rule:
22
Note that the definition above makes sense because the sum overa+f3= l' is fillite for any fixedl'EZ~).Moreover, if we sett;
=
t", then, for any aEZ~),t·~
II t:"',
iEsuppo
so k[[tili E1]Jcontains the algebra of polynomials k[t;li EI]as a dense subalgebra
Finally, k[[t;li EJ)Jis a pro-finite topological algebra with a fundamental system of neighbourhoods of 0 consisting of the ideals
In gelleral, any pro-finite topological algebra has a fundamental system of neighbourhoods of 0 consisting ofideals[14, Section1.2.71.
0.5 One Fact from Descent Theory
We will need the following standard descent theory lemma (see e.g [36, Chap- ter 17]). Recall that if L/k is a (possibly infinite) Galois field extension, then we can dcfme theKrtlll topologyon the Galois group E
=
Gal(Llk) by taking as a fundamental system of neighbourhoods of 1 all the centralizer subgroups of finite sllbextellsions. Then E becomes a c:ompa<:t Hausdorff topological group and we recover in the general case the classical bijection between subgroups and subfields (that holds for finiteLlk) if we restrictDEFINITIOIVS AND BASIC FACTS 23 our attention ollly toclosedsubgroups(see e.g. [22]). Moreover, any open subgroup ofE is of finite iIldcx, so E is a pro-finite group.
Lemma0.5.1.Let Llk be a GaloM field extension,L:
=
Gal(Llk}. Let V be a vector space over L endowetl with a continuous semilinear E-action, i.e.s(V+w)=s(v)+8(W).S(AV)=S(A)S{V), 'Is E E, v, w EV,A E L, Midthe centralizer of any vector' in Y is an open subgroup ofE. Then
y=y E0k L ,
whereyE C Vis the set olE-invariants. Moreover,yEinherits any algebraic structure defined onVby E-invariant L-multilinear m(lps.
•
Chapter 1
Identities of Coalgebras
1.1 Coalgebras with a Polynomial Identity
Itseemsnatural to define a polynomial identity for coa!gcbras using their duality with algebras, for which this notion is quite well-known (see Section 0.3). The following definition was introduced by the author in [19].
Definition1.1.1.Let C be a coalgebra over a field k,F(X1 ,..,Xn )a polynomial in n noncommuting variables with coefficients in k. We say that F
=
0 (or justF)is an identity for the coalgcbra0, ifitis all identity for the dual algebra C·Using duality, we immediatelyobsen"ethatifa coalgebra C satisfies some idcntiw, then any subcoalgebra and any factorcoalgebra of C satisfies this identity.Ifa family of coalgebras satisfies some identity, then their direct sum satisfies this identity
Since any coalgebra C is the sum of its finite-dimensional subcoalgebras (see Theorem 0.1.8), in order to prove thatF=0 holds for C, it is sufficient
24
CHAPTER1. lDENTI'l'IES OF COA.LGEBRAS 25 to verify tbis identity on finite-dimensional subcoalgebras of C. We will use this observation later.
Now we want to give an intrinsic definition when a coalgebra satisfiesall
identity, i.e. a definition that would not use the dual algebra. Unfortunately, it works well only for multilinear identities (which is sufficient in the case of char k "" 0 because of Theurem 0.3.2)
A multilinear polynomial of degree n has the form:
F(X1,..,Xn )
= L
A"X"(l)".X..(n),"ES~
whereSn is the group of permutations and '\". Ek.It will be convenient to identifyF with the clementL"ES~'\".11"of the group algebrakSn
For any v(.'Ctor spaceV,there are natural right and left actions of 8n on
V~;In:
(VI0 ..0v,,)·1I"
11"'(v10 .. 0vn)
Then the fact that an algebra A satisfies a multilinear identity F
=
0 can be written as follows:mn(A&n.F)
=
0,where1nn ;A®n-+A is the multiplication of A. The following dual definition for coalgebras is due to Yu.Bahturin.
Definition 1.1.2.Let C be a coalgebra,F ""L"e"~>."X".(I) . X,,(n)a multilinear polynomial. We say that C satisfies the identityF
=
0 if where 6" : C-+C(i:n is the comultiplicatioll of C.CHAPTER1. IDENTITIES OF COALGEBRAS 26 Soa multilinear identity of degreencan be viewedas asort of symmetry condition on. the tensors .6."c, for all c E C.
Proposition 1.1.3.DefinitiortS1.1.Jand1.1.2 are equivalent for multilJnear identities.
Proof. USiJlg the sigma notation.6.nc=2:C(l}08' ...0c{n),C EC,we have:
F·(.6. nc)=
L:),.-
L:C('--'(l)}€I .. \8lC('--'(n»,"ESn hence, for all'PI, .. ,'P"EC',
(rp I0 .. \81'Pn,P·(A"c)) I:>'.I:(~"~''''''lJ) "('."('''''.lJ)
llESft
I:>'.I:(~.,,,,,<,,).. (,.(.),,<.,1
llESn
L:),,,(rp,,(l) ..rpll(n),Cl 1rESn
Therefore,F (.6.nC)
=
0iffthe identityF('PI," ,'Pn)=
0 holds for all•
Using the sigma notation as in the proof, the fact that a coaigcbra C satisfies a multilinear identity can be written as follows:
2=: ),,,
L:X(ll-'(I})€I ..<9x(ll-'(n)=0, Vx E C.llES~
For example, cocomffiutativity can be expressed like this'
L:x(l)€IX(2) - L:X(2)€IX(I)
=
O.CHAPTER1. IDENTITIESOFCOALGEBRAS 27 By Definition1.1.1,any finite·dimcnsionaJ coalgebra C satisfies the stall- dard identity
E
(_1)rE
x(r-1(1}) 119 .. 119 x(r-1(nn=
0, rESRfor anyn>dime.
The following propositionpro~'idesa way of constructing infinite-dimen- sional PI-coalgcbras(i.e.coalgebras witil a nontrivial identity).
Proposition 1.1.4.Ifanalgebra A satisfies the identJtyF
=
0,then so does UICcoa/gebraAO.The converse holds if A is residually finite-dimensional.Proof. For the first assertion, it sufliccs to prove that F=0 is an identity for any finite-dimensional subcoalgebra D CAO•SetI
=
DJ.. This is an ideal of finite codimension inA.Since D is finite-dimensional, 1.1.=
D,and so we have D2!'(A/1)°=
(A/I)*,hellce D* ""'"A/[ satisfies F=
0Conversely, ifAis residuatly finite-dimensional, Le. the intersection of the ideals of finite codimension inAis 0, thenACAO•.But the algebraAO*
satisfiesF
=
0 since so does the coalgebraAO.1.2 Free Coalgebras
•
A polynomial identityF(Xj, . ,Xn) of an (associative) algebra lIIay be con- sidered as an element of the free (associati\'e) algebra with n generators, i.e.
the tensor algebraT(V},whereV= (XI, .. ,Xn )(recall that we assume that algebras have the unit element).
In order to make a link between identities of coalgebras and free coalge- bras, we first need to define the latter. Free coalgebras (whidl more precisely
CHAPTERJ. IDENTITIES OF COA.LGEBRAS 28 should be called "cofreecoalgebra.s~)were introduced by M.Sweedler ill 132].
They are defined by the following universal property, which is dual to the universal property of tensor algebras.
Definition 1.2.1. Let V be a vector space, C a coalgebra,(J :C- tVa linear map. The pair(C,9) iscalled afree coalgebm of Vif, for any coalgebra D and a linear maprp :D- tV,there exists a unique coalgebra map.p :D- tC completing the commutative diagram:
D----<t'---C . " /8
V
By a standard argument, if a free coalgebra ofVexisL~,it is unique up to a uniquely definedisomorphi~m.We will denote it bycT(V). Itis shown in [321 thatcT(V)exists for anyV,but we will follow a morc explicit construction of R.Block and P.Leroux [10]. First we introduce the generalized fillite dual.
Definition 1.2.2.Let A
=
EBn?:OA~be a graded algebra, V a vector space.Let Hom(A, T(V)) denote the space of all graded linear functions of degree ofromAto the tensor algebraT(V)=EB~?:o1"'(V),i.e. all linear functions f:A- tT(V)such that ifa E A.n ,thenf(a) is a tensor of degree11.Wewill caJlfE Hom(A,T(V)) representative if there exists a finite family {g;,h;}
of elements of l-Iom(A,T(V))such that
flab)~:[9,(a)h;(b), Va,bEA, (1.2.1) where the multiplication on the right-hand side is the tensor product inT(V).
Since we will later consider elementsofT(V) 0T(V),we reserve the symbol
o
for the"outer~tensor product and simply writeVI ••v~for tile elementCHAPTER1. IDENTITIES OF COALGEBRAS 29 Vj<81 ..oSlvnE l'(V). The set of all representative functions A-+1'(\1)will be denoted by A
v.
Itfollows thatiffEA\.-, then the tensor
2:i
9,011,; is uniquely determined by (1.2.1). We definef:!"f ::2:;
9, 0h;,and it turns out that6fEAv
<81At"and(A'V,f:!,,) is a coalgebra with counit c(f}=f(1) [10, Lemma 1 and The- orem 1]. IfV ::k, we recover the usual finite dual coalgebra AOof the (underlying ungraded) algebraA
Ifwe now specifyA
=
1'(W)(graded by degree), for some \'ector space W,then there is a natural linear map0: T{W)v--4Hom(W,V)which sends fE1'(W)Vto its restriction toW = 1'1(W).Theorem 1.2.3 (Theorem 2' in [10]).Let V andIV bevector spaces.
TIIefl(T(W)'V,O)defined aboveis(arealization of) thefreecoalgebra of the spaceHom(W, V).Moreover, 1fDis a coalgebra andiIJ D-+Hom(W, V) i.,a linear map, then the lifting ofiIJto a coalgebra map4> :D-+T(W)'Vis givenby
~(d)l '(d),
<fl(d)w rp(d)w, Vw EW :: 1'1(W),and
lI>(d)z (Lli?(d(l))<8I ..o<P(d(R)))z, ';/zET'(W),ifn>1.
•
III particular, if we setV ::k, we see that1'{W)Ois the free coalgebra ofW·. This is a result of M.Sweedlcr originally used to prove the existence of free coalgebras.Italso sheds light on the nature of identities of a coal- gebra. LetF(X1,..,X,,) be an associative polynomial in n \'ariables, set
CHAPTER1. IDENTITIESOFCo.4.LGEBRAS 30 W = (Xl' ., Xn ),SOFET(W).Then the free coalgebracT(W") = T(W)O is a subspace ofT(W)"containingT(W')(sec Remark 1.2.4 below). More- over,T(W)"has a natural topology of a dnal space (see Definition0.4.1), andT(W)can be recovered as the space of all continuous linear fuuctions on T(W)",SinceT(W")is dense inT(W)",so iscT(W')and hence the spaces of continuous linear functions onT(W)"and oncT(W')(with topology in- herited fromT(W)")are in one-to-one correspondence, Thus we conclude thatT(W)is the space of contiI1UOUS linear functions oncT(W")and so poly- nomial identities inXl, . ,Xncan be viewed as continuous linear functions on the free coalgebra of the space(Xl,'" ,Xn )".
On the other hand, if we setW
=
k in Theorem 1.2.3, we obtain that k[tl'V is the free coalgebra of V. This gives a rather explicit. construction of cT(V)as follows. DClwteT(V)the completion of the tensor algebra1'(V), i.e, the algebra of all infinite fOflllal sumsZ=Zo+z\+...,whereZ;Er(V).The topology onT(V)is defined by a fundamental system of neighbourhoods of 0 consisting of the sets
Fnj'(V)
=
{z ET(V)IZ;=
OV'i<n}.Theil an elementIE Hom(k[t],T(V))call be identified with the formal sum 10+Ii+ .. ",whereIi=I(t;}E'1"(V),and so cT(V} be<:omcs a subspace ofT(V). Upon this identification, the canonical map0 cT(V) --t Vjust sends the sumIa+ Ii+ .".to its degree1 componentIiE Tl(V)=V,and the formulas of Theorem 1.2.3 for the lifting of a linear maplp D--tVto
CHAPTER 1. IDENTITIES OF COALGEBRAS a coalgebra map<II :D ....; cT(V) become:
<I!(d), 'Id),
<fl(dh :p(d),and
If>(d)"
L
1p(d(l))0 ··0:p(d(.. ))ifn>1.31
Moreover, an explicit formula for the cOlTlultiplication ofcT(V) can be obtained as follows (see [9, Sections 1 and 2]). LetV denote the continu- ous linear function fromT(V)toT(V)0T(V)defined by its action on the monomials'
V(Vj. v,,)=tV! ... v,0v;+l
;:0
\tVl,.,V..EV,n=O,l, ..
(1.2.2) Then an clementf ET(V)belongs tocTW)iffVflies in t·he subspace T(V) 0T(V)cT(V)0T(V),in which case6f
=
VI,i.e. the cumultiplica- tion ofcT(V) is just the restriction of VoncT(\I). We also see from here thatT(V) C cT(V). In particular, this implies that the canonical map ()is surjective. The counit ofcT(V) just sends the sum10+I)+ ..to its degreeo
component10E'j">4\V)=k.Remark 1.2.4. Assuming the spaceV finite-dimensional, set W
=
V'.Then the above construction ofcT(V) C f(V) agrees with the construction of M.5weedler which realizescT(W')as the subspaceT(W)OofT(W)'= T(V).III particular,T(W)OcontainsT(W·).
Remark 1.2.5. R.Block also proves in [9] a number of interesting properties ofcT(V) which we will not usc here. But one thing should be mentioned.
since it illustrates the duality with algebras. Namely, there is a natural
CHAPTER1. IDENTITIES OF COALGEBRAS 32 multiplication
* :
r.T(V) 0 cT(V)-+cT(V) which is the lifting of the linear map00f;+f:0(j :cT(V)€IcT(V)-+V, so cT(V) has a structure of a commutative Hopf algebra, with the antipode induced by -8 :T(V)-+V - dually to the fact the free algebraT{V) has a natural structure of a co(.'Ommutative Hopf algebra defined byV-+T(V)0T{V): v-+v01+10v, with antipode induced byV-+T(V) v-+-v (this is the same Hopf algebra structure as the olle coming from the fact thatT(V)is the universal envelope of the free Lie algebra generated by the spaceV).We will return tothe multiplication*
in Section 1.5, where we will see that itisin fact the so called "shuffleproduct~.To conclude this section, let us introduce the notion of a cogenerating map for coalgebras, whichisthe dual of a generating set (or, more precisely, space) for algebra.<;. LetA be all algehra, V a vector space. Suppose we have a linear IllapIf) :V--tA,then the imageIf)(V)generatesAas all algebra iff
I:
(m.0~"")V0'~A,"2:0
where1n" :A0fl-+A is the multiplication of A (with mt
=
idA and mo=
u, the unit map). The formal dual of this statement is the following:Definition 1.2.6. LetCbe a coalgebra,Va vector space. We will call a linear mapIf) :C--tVcogeneratingif
nK"(~·'o",) ~O
"2:0
A generating set in all algebraA allows us to represent A as a fador of a free algebra. Dually, (.'Ogenerating maps for a coalgebra C correspond to the imbeddings of C into free coaigebras.
CHAPTER1. IDENTITIES OF COALGEBRAS 33 Proposition 1.2.7.Del C be a coa/gebra, V(lvector space,1p :C-tV a linear map. Then the induced coa/gebra mapIf.I :C--+c'f(V)isinjective if!
'Pis cogcncrating.
Proof. Recall from the explicit construction ofcT(V) that 1>(d)" =:
Llp(d(l)0 ... 0'P(d(nj),dEC(with the convention that the right-hand
side meallsI.p(d)forn =: 1 ande:(d)forn =: 0). In otller words,<f.>(d)"=:
('P@"otl.,,)d,hencedE Ker1> iff (y;@notl.,,)d=:0,for alln. • III particular, any coalgebra can be imbedded into a free coalgebra (take Ii=C,thenid: C- tVis obviously a cogcnerating map)
1.3 Varieties of Coalgebras and Theorem of Birkhoff
In this section we assume the field k infinite.
First we briefly recall the situation that we have for algebras LetF denote the free algebra in coulltably mall)' generators, i.e.F =: T(V), where V=: (Xt,X2 , ••. ). Then any polynomial identity, no malter in how many variables, can be viewed as an element ofF.
LetA be an algebra, then the set I(A) of all identities satisfied by A is an ideal ofF,invariant under any endomorphism ofF.
Definition 1.3.1. An idealJ ofFis called aT-idealifcr(J)CJ, for any endomorphism cr ofF,or, equivalently, iff3(J)=0,for any algebra map f3:F-tF/J (the equivalence follows from the univcTlial property of F)
CHAPTER1. IDENTITIESOFCo.4.LGEBRAS 34 Definition 1.3.2. LetShe a subsetof:F.Thcvariety of algebrasdefined by Sisthe class Var(S) of all algebras that satisfy each identity from the set
Then varieties of algebras arcillone-to-onc correspondence with T-ideals as follows.If2{is a variety defined by someSc:F,then the setI(21) of all identities satisfied by each algebra from 21 is the T-ideal generated by S (Le.
the smallest T-ideal containingS).In other words, the T-ideal generated by S consists of all possible consequences of the identities fromS.Therefore, if Jc:Fis already a T-ideal, then for the variety of algebras 21
=
Var(J) we haveI(21)=
J.Conversely, ifQlis a variety, then dearly 21=
Var(I(21)).Varieties of algebras can be dlaracterized by the following theorem [8].
Theorem 1.3.3 (Birkhoff).Let21.be a nonempty class of algebras. Then21.
isa variety (i.e.isdefined by identities) iff'J.isdosed under isomorphisms, Mjbalgebm.~,faetoralgebms, and directprodue~. •
Now we turn our attention to coalgebras. By analogy with algebras, it is natural to give the following definition.
Definition 1.3.4. Let S be a subset of:F. Thevariety of coalgebmsdefined bySis the class eVartS) of all coalgebras that- satisfy each identity fromS.
Here we also have a one-ta-one correspolldcllce between varieties and T-ideals. Since by definition the setI(C) of identities of a coalgebra 0 is thc same as the set of identities of the algebra0·)I(O) is aT-ideal.
Consequently, ifl!
=
cVar(S), then the set I(I!) of all identities satisfied by l!is a T-ideal containingS.Itis not immediately obvious whyI(I!:) shouldCHAPTER1. IDENTITIES OF COALGEBR.4.S
be the smallest such T-ideal, but we will prove this in a moment (and we will need the assumption that k is infinite). \I'lith this fact at hand, the mapsI and cVar establish the desired one-to-one correspondence as in the case of algebras.
Proposition 1.3.5.LetItbe the variety of coalgebrns defined by a set of identitiesS.Then the T -idealI(It)of identities ofItisgenerated bySas a T -ideal. In other words, the consequences of the system of identitIes S arc the same for coulgebras as they are jor algebras.
Proof. First of all, since the base field k is infinite, any 7'-idealJis graded (sec Theorem 0.3.2). It follov.'s that the algebraFIJ is residually finite-dimensional. Indeed, for allYF(XI, ... ,X,,)~J,we need to find an ideal of finite codimensionJ' :> J such thatFf/;J'. SinceJis graded, we can setJ'equal to the ideal generated byJ,X,,+J,X"+2, ..and by all monomials inXl, .. , X"of degreed+1, wheredis the maximum degree of monomials occuring inF.
Now letJ be an arbitrary T-ideal containing our setS.By Proposition 1.1.4, the coalgebra D=(FIJ)O satisfies the same identities as the algebra FIJ, soI(D)=J.SinceScJ,Dis in the variety I!:, henceJ=I(D):>
I(It).Therefore,I(It)is the smallest T-ideal cOlltainingS. • Surprisingly enough, the analog of Theorem 1.3.3 does not hold for coal- gebras. Obviously, any variety of coalgebras is closed under isomorphisms, subcoalgebl'as, factorcoaIgebras, and direct sums. HO\\'e\'er, not every such class is a variety.
Example 1.3.6. The classGrpof all coalgebras spanned by group-like ele- ments is closed under the four operations just listed, but itisnot a variety.
CHAPTER1. TDENTITIESOFCOALGEBRAS 36 Proof. First, if a coalgebra Cisrepresented as a direct sum of coalgebras:
C
=
EBiCj, then any subcoalgebraDeChas the form D=EB,Di ,where D;=D nCi .This is the fact dual to the following statement for algebras:ifA
=Di
Ai>then any idealJ CAhas the formJ=n,
J" where J; is the projection ofJ to Ai<It follows that if C is spanned by group-like elemcnts (which are nccessarily linearly independent), then any subcoalgebra of C is just a span of a subset of these group-like elements. Second, if C is spanned by group-like elements, then any homomorphic image of C is spanned by the images of these elemcnts, which are eithcr group-like or zero. Obviously, the class Grp is also closed under isomorphisms and direct sumsGrp is not a variety, because it is properly contained in the variety Cocomm of all coeornmutative algebras, which dOL'S not have ally proper subvarieties other thanto}.The latter is the case since any T-idealcontain~
ing the idcntityXlX 2 - X 2Xjis either geucrated by it or is the whole:F by
Corollary 0.3.3.
•
Example 1.3.7. The classPntof all pointed eoalgebras is closed under the four opcrations listed above, but it is not a variety
Proof. Recall that a coalgcbraiscalled pointed if all its simple subcoalge- bras are one-dimcllliional or, equivalently, its coradical is spanned by group- like clements. Thus Put is obviously closed under isomorphisms and sub- coalgcbras. Further, by Corollary 0.1.13, a homomorphic image of a pointed coalgebra is pointed. Finally, [25, Lemma 5.6.2(I)J says that ifC
=
EiG;, where GjC G are subcoalgebras, then allY simple subcoalgcbra of C lies in one of the0;, hence a sum of pointed coalgebras is pointedExample 1.5.1 in Section 1.5 (discussing Taft's algebras) implies thatPnt
CHAPTER1. IDENTITIES OFCOAl~GEBR.AS 37 does not satisfy any nontrivial identity. HencePnt is a proper subclass of the ....arietyCoaly of all coalgebras, which is not contained in any proper
subvariety.
•
Definition 1.3.8. We will use the termpseudo-varietyfor any nonempty c1lk~sof coalgebras closed under isomorphisms, subcoalgebras, factorcoalge- bras, and direct sums.
ThnsOrp and Pnt are pseudo-varieties which are not varieties. We have shown that varieties of coalgebras are in one-to-olle correspondence withT- ideals in the free algebraF in couurably many generators XltX2 , • •To characterize pseudo-varieties ill a similar manner, we will need the objects dual to T-ideals. Letc:F=cT(V), where V=(X1,XZ, .. .). Loosely speak- ing,c:Fis the free coalgebra "in conutably many cogencrators" . Definition 1.3.9. A subcoalgcbraLCcFis called aT-subcoalgebmif a(L) C L, for allY endomorphism0'ofc:F,or, equivalently, if,8(L)cL, for any coalgebra map/3 :L ...c:F(the equivalence follows from the univer- sal property ofc:F).
Then pseudo-varieties of coalgebras are in a one-to-one correspondence with T-subcoalgebras as follows. We associate with a pseudo-varietyI[the largest snbcoalgcbraLcc:Fbelonging to<t(the sum of all such subcoalge- bras, which belongs toI..':becauseI[is closed under direct sums and factors).
SinceI[is closed under homomorphic images,Lwill be a T-subcoalgcbra.
Conversely, we associate with a T-subcoalgcbraLcc:Fthe classI[con- sisting of all coalgebrasDsuch that, for allY coalgebra map "( : D- tc:F,
"((D)cL.Obviously, ([ is closed under isomorphisms, factors and direct
CHAPTERJ. TDENTITJES OF COALGEBRAS 38 sums, It is also closed under subcoalgcbras, because coalgebra maps to
cr
can always be e."{tended from subcoalgcbras by the universal property of
cr.
Two more checks are necessary.
Firstly, letLbe a T-subcoalgebra,I[the pseudo-variet.y associated with L,andL'the T-subcoalgcbra associated with1[. SinceL'belongs to1[, then considering the inclusion mapL'"-+r, we see thatL'cL by the construction ofot. Convcrsely, using the definition of a T-eoalgebra,\\"C
conclude that L also belongs toot, but then LcL'sinceL'hithe largest subcoalgebra with this property. SoL = U.
Secondly, letot be a pseudo-variety,Lthe T-subcoalgebra associated wit.h 1[,andIt'the psendo-variety associated with L, If a coalgebra D belongs to
<!:, then for any coalgebra map, 0--tcr,,(0)cLsinceot is closed under homomorphic images andLis the largest subcoalgebra ofc:Fbelonging toot, Therefore, 0 is in1['and we proved thatI[c1['.Conversely, if a coalgebra 0 belongs toot', we wanttoprove that D must be in<!: and soIt C1[.To ihis end, observe that it suffices to prove that any finitt-'-dimcnsional subcoalgebra of0 lies inot, because 0 is a sum of such snbcoalgebrali andI[is closed under sums. So we may assume 0 finite-dimensionaL Then there isallinjective linear map;o:0--tV
=
(XJ,Xz, .. .), which can be lifted to a coalgebra map l]:>:D--tc:F,necessarily also illjccti\'e. Since <I>(D) CLby the definition of the classIt, we conclude thatDis isomorphic to a subcoalgebra ofL,hence D is inot. This completes the proof of the desired one-to-one correspondence.To conclude this section, we will give a characterization of varieties of coalgebras among pseudo-varieties. The following replacement of Birkhoff's theorem says that a pseudo-variety is a variety iff it is closed under some sort
CHAPTER 1. IDENTITIES OF COALGEBRAS of "completion".
39
Theorem 1.3.10.A 1l0nempty class of coalgebras <1: is a variety iff itis closed lmder~somorphisms,stllJcoajgelJrns, jactorcoalgelJrns, and direct sums, a1ld in addition,for any coalgelJraCfrom [ and a1ly sulJalgelJraAcC', A° IJelongs toIt.
Remark 1.3.11. In the theorem above, it suffices to cOllsider only subal- gcbrasA C C' that are dense in the topology of the dual space. In this case, C imbeds intoAO,so the latter can be regarded, loosely speaking, as
"completions" of C (not ill the topological sense: C has no topology).
Beforewe can prove Theorem 1.3.10, lYe will oeM the following useful characterization of the T-idealI(C)of identities of a coalgebra C. This lemma is a dualizatioll of the statement: the T-idealI(A) of identities of an algebraAis equal to the intersection of the kernels of all algebra maps :F --rA. Recall the notation of Lemma 0.1.5.
Lemma 1.3.12.LetCIJe a coalgebra. Denote lJy L the sum of the images ofallcoalgebra mapsC--+P. Then I(C)
=
Ll.Proof.Recall from Section 0.1 that the functor ()0 is the right adjoint of(j", i.e. for any algebraAand coalgebra C, the sets of homomorphisms Alg(A,C')and Coalg(C,AO)are in a one-to-one correspondence. Namely, the following are the inverse bijections constructed by M.Swcedler[321:
<1>:Alg(A,C')--+OJalg(C,AO) sendingt3to the composite C'-tC·Ot;.AO, and
11': Coalg(C,AO)--rAlg(A,C') sending0'to the compositeA --+N'~C'.
